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Topological and geometric inference of data
The overarching problem under consideration is to determine the structure
of the subspace on which a distribution is supported, given
only a finite noisy sample thereof. The special case in
which the subspace is an embedded manifold is given particular
attention owing to its conceptual elegance, and asymptotic bounds are
obtained on the admissible level of noise such that the
manifold can be recovered up to homotopy equivalence.
Attention is turned on how to accomplish this in practice.
Following ideas from topological data analysis, simplicial complexes are used
as discrete analogues of spaces suitable for computation. By utilising
the prior assumption that the data lie on a manifold, topologically
inspired techniques are proposed for refining the simplicial complex
to better approximate this manifold. This is applied to the
problem of nonlinear dimensionality reduction and found to improve accuracy
of reconstructing several synthetic and real-world datasets.
The second chapter focuses on extending this work to the
case where the ambient space is non-Euclidean. The interfaces between
topological data analysis, functional data analysis, and shape analysis
are thoroughly explored. Lipschitz bounds are proved which relate several
metrics on the space of positive semidefinite matrices; they are then
interpreted in the context of topological data analysis. This is
applied to diffusion tensor imaging and phonology.
The final chapter explores the case where the points are
non-uniformly distributed over the embedded subspace. In particular, a method
is proposed to overcome the shortcomings of witness complex construction
when there are large deviations in the density. The theory
of multidimensional persistence is leveraged to provide a succinct setting
in which the structure of the data can be interpreted
as a generalised stratified space.EPSR
Invariant means on Abelian groups capture complementability of Banach spaces in their second duals
Let be a Banach space. Then is complemented in the bidual if
and only if there exists an invariant mean with
respect to a free Abelian group of rank equal to the cardinality of
, and this happens if and only if there exists an invariant mean with
respect to the additive group of . This improves upon previous results
due to Bustos Domecq =and the second-named author, where certain idempotent
semigroups of cardinality equal to the cardinality of were considered,
and answers a question of J.M.F. Castillo (private communication). En route to
the proof of the main result, we endow the family of all finite-dimensional
subspaces of an infinite-dimensional vector space with a structure of a free
commutative monoid with the property that the product of two subspaces contains
the respective subspaces, which is possibly of interest in itself.Comment: 12 pp., accepted for publication in Studia Mathematic